Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyFTC-IITheorem: Fundamental Theorem of Calculus - Part II [FTC-II]Check Conditions of FTC-IIActivity 16CC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**Objective 01–01**: I can state the*Fundamental Theorem of Calculus*.**Objective 01–02**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–03**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–04**: I can determine the properties of an area function using*FTC-I*and*Calculus I*.**Objective 01–05**: I can evaluate indefinite integrals using the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–06**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–07**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **Objective 01–08**: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.**Objective 01–09**: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Let

is the black graph. Is$f(t)$ continuous on$f(t)$ ? Explain.$[1,12]$ **NOTE**: A function is continuous on an interval**IF**:there are

**NO***holes*in the graph;there are

**NO***jump discontinuities*in the graph;**and**there are

**NO***infinite discontinuities*in the graph.

**NOTE**: This DESMOS activity will help you understand the **Fundamental Theorem of Calculus - Part II (FTC-II)**. For a good proof (which is beyond the expectation for this course) of both *FTC-I* and *FTC-II*, see https://math.berkeley.edu/~peyam/Math1AFa10/Proof of the FTC.pdf.

**Procedure**:

Let

Move the slider for the parameter

[Box 3] to a value of$a$ .$a=12$ Identify the

*net area*[the value listed in Box 2]*under* [black curve].$f(t)$ From the graph, identify the value of the antiderivative

function [green curve] when$F(x)$ .$a=12$ From the graph, identify the value of the antiderivative

function [green curve] when$F(x)$ .$a=1$ Evaluate

and compare to the$F(12)-F(1)$ *net area*under [black curve]. What do you notice?$f(t)$

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Created**: Tuesday, 25 August 2020 03:53 EDT
**Last Modified**: Monday, 30 May 2022 - 14:32 (EDT)